We exhibit examples of slope-stable and modular vector bundles on a hyperkähler manifold of \(K3^{[2]}\)-type. These are obtained by performing standard linear algebra constructions on the examples studied by O’Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic 4-fold and the Debarre-Voisin hyperkähler. Interestingly enough, these constructions are almost never infinitesimally rigid, and more precisely we show how to get (infinitely many) 20 and 40 dimensional families. This is a joint work with Claudio Onorati. Time permitting, I will also present a work in progress with Alessandro D'Andrea and Claudio Onorati on a connection between discriminants of vector bundles on smooth and projective varieties and representation theory of \(\mathrm{GL}(n)\).